Theorem bj-cbv3ta 34110

Description: Closed form of cbv3 2415. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-cbv3ta	⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Proof of Theorem bj-cbv3ta

Step	Hyp	Ref	Expression
1		bj-spimt2 34109	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
2	1	imp 409	. . . . 5 ⊢ ((∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ (∃𝑥𝜓 → 𝜓)) → (∀𝑥𝜑 → 𝜓))
3	2	alanimi 1817	. . . 4 ⊢ ((∀𝑦∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∀𝑦(∃𝑥𝜓 → 𝜓)) → ∀𝑦(∀𝑥𝜑 → 𝜓))
4		bj-hbalt 34017	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
5		sylgt 1822	. . . 4 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
6	3, 4, 5	syl2im 40	. . 3 ⊢ ((∀𝑦∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∀𝑦(∃𝑥𝜓 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))
7	6	expimpd 456	. 2 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
8	7	alcoms 2162	1 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  bj-cbv3tb  34111